Amplitude modulation with large carrier

Perhaps the simplest form of (analog) transmission system
modulates the message signal by a high frequency carrier in a two-stepprocedure: multiply the message by the carrier, then
add the product to the carrier.At the receiver, the message can be demodulated
by extracting the envelope of the received signal.

Consider the transmitted/modulated signal

v(t)=Acw(t)cos(2πfct)+Accos(2πfct)=Ac(w(t)+1)cos(2πfct)

diagrammed in
[link] .
The process of multiplying the signal in timeby a (co)sinusoid is called
mixing .
This can be rewritten in the frequency domainby mimicking the development from
[link] to
[link] .
Using the convolution property of Fourier Transforms
[link] , the transform of
v(t) is

V(f)=F{Ac(w(t)+1)cos(2πfct)}=AcF{(w(t)+1)}*F{cos(2πfct)}.

The spectra of
F{w(t)+1} and
|V(f)| are
sketched in
[link] (a) and (b).
The vertical arrows in (b) represent the transform of the cosine carrierat frequency
fc (i.e.,
a pair of delta functions at
±fc ) and
the scaling by
Ac2 is indicated next to the arrowheads.

A communications system using amplitude modulation
with a large carrier. Inthe transmitter (a),
the message signal
w(t) is modulated by a carrier wave at frequency
fc and then added to the carrier to give the transmitted
signal
v(t) . In (b), the received
signal is passed through an envelope detector consistingof an absolute value nonlinearity followed by a lowpass
filter. When all goes well, the output
m(t) of
the receiver is approximately equal to theoriginal message.Spectra of the signals in the large carrier amplitude modulation
system of
[link] .
Lowpass filtering (d) gives a scaled version of (a).

If
w(t)≥-1 , the envelope of
v(t) is the same as
w(t) and an
envelope detector can be used as a demodulator.
One way to find the envelope of a signal is to lowpass filterthe absolute value. To see this analytically, observe that

where the absolute value can be removed from
w(t)+1 because
w(t)+1>0 (by assumption).
The spectrum of
F{|cos(2πfct|} ,
shown in
[link] (c), may be familiar from
Exercise
[link] .
Accordingly,
F{|v(t)|} is the convolution
shown in
[link] (d). Low pass filtering
this returns
w(t)+1 , which is the envelope of
v(t) offset by the constant one.

An example is given in the following M
atlab program. The
“message” signal is a sinusoid with a drift in the DC offset,and the carrier wave is at a much higher frequency.

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Tarell

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Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

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